List the possible rational zeros. $$ n(x)=25 x^{5}+8 x^{3}-2 x^{2}+10 $$ If there is more than one answer, separate them with commas. Select "None" if applicable. The possible rational zeros of $$ n(x): \square $$

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The Rational Zeros Theorem tells us that any rational zero of the polynomial

$$
n(x)=25x^{5}+8x^{3}-2x^{2}+10
$$

must be of the form

$$
\frac{p}{q}
$$

where $$ p $$ is a factor of the constant term $$ 10 $$ and $$ q $$ is a factor of the leading coefficient $$ 25 $$.

The factors of $$ 10 $$ are:

$$
\pm 1, \pm 2, \pm 5, \pm 10
$$

The factors of $$ 25 $$ are:

$$
\pm 1, \pm 5, \pm 25
$$

Thus, the possible rational zeros are all numbers of the form

$$
\frac{\text{factor of }10}{\text{factor of }25}
$$

This gives:

$$
\pm \frac{1}{1}, \pm \frac{2}{1}, \pm \frac{5}{1}, \pm \frac{10}{1}, \pm \frac{1}{5}, \pm \frac{2}{5}, \pm \frac{5}{5}, \pm \frac{10}{5}, \pm \frac{1}{25}, \pm \frac{2}{25}, \pm \frac{5}{25}, \pm \frac{10}{25}
$$

We simplify the fractions:

$$
\pm 1, \pm 2, \pm 5, \pm 10, \pm \frac{1}{5}, \pm \frac{2}{5}, \pm 1 \, (\text{from } \frac{5}{5}), \pm 2 \, (\text{from } \frac{10}{5}), \pm \frac{1}{25}, \pm \frac{2}{25}, \pm \frac{1}{5} \, (\text{from } \frac{5}{25}), \pm \frac{2}{5} \, (\text{from } \frac{10}{25})
$$

After removing duplicates, the distinct possible rational zeros are:

$$
\pm 1, \pm 2, \pm 5, \pm 10, \pm \frac{1}{5}, \pm \frac{2}{5}, \pm \frac{1}{25}, \pm \frac{2}{25}
$$

Thus, the answer is:

$$
\pm 1, \pm 2, \pm 5, \pm 10, \pm \frac{1}{5}, \pm \frac{2}{5}, \pm \frac{1}{25}, \pm \frac{2}{25}
$$
The possible rational zeros of $$ n(x) $$ are $$ \pm1, \pm2, \pm5, \pm10, \pm\frac{1}{5}, \pm\frac{2}{5}, \pm\frac{1}{25}, \pm\frac{2}{25} $$.

List the possible rational zeros. \[ n(x)=25 x^{5}+8 x^{3}-2 x^{2}+10 \] If there is more than one answer, separate them with commas. Select "None" if applicable. The possible rational zeros of \( n(x): \square \)

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